Suppose I wanted to obtain the default probability of a firm in two ways, and plot the difference in the two. It's straightforward to pull default probabilities (in a loose sense) from CDS spreads, but is there a way to do this with equity and/or equity option data?

Merton gives some framework for this, assuming zero recovery for equity and i've seen a few extensions to account for non-zero equity recovery.

Bottom line question is: can I find an equation or estimation of default probability using only equity market data in the same way I can "translate" cash bond spreads into a CDS-equivalent for purposes of calculating a basis?

**EDIT**

I have a Carr and Wu (2011) method that is easy to replicate for turning put option data into default probabilities. My more technical question is how does mis-estimation of the recovery rate on the CDS side of the trade impact the equity side of things?

What are some of the biases relating to recovery rate that I'm introducing by calculating a measure of, say,

CDS_p(default) - Put_p(default)?

"These metaphors and similes aint similar to them, not at all." -Eminem

Merton says that a company defaults when its equity is worthless. So if you go by Merton, you are estimating when the share price will hit 0.

Carr and Wu have replaced that with share price hitting some crash put strikes above zero.

That is all OK in theory.

In reality, crash puts are a mug's game. Nobody makes money buying crash puts, and in the long run nobody makes money selling them either. Plus, liquidity in crash strikes is non-existent and information content in prices is zero.

"People say nothing's impossible, but I do nothing every day" --Winnie The Pooh

I guess the Merton model is the canonical way to go. But... you have to estimate the default barrier (how do you account for short term debt? Moody's KMV used all long term debt + 50% short term debt iirc) and what would happen if the default barrier is crossed between two coupon dates (if you can't pay but you don't have to there is no default, no? Almost Zen like...)? Finally I would guess that there are some pricing differences between the two markets and that prices don't always fit exactly. Ages ago there was a product called Equity Default Swap, which was basically a one-touch put in disguise, that should mimic CDS in equity land but it never gained any traction afaik.

Have great deal of respect for Einchcomb, although I'm always skeptical of big "arbitrage" models; market disruptions tend to overshadow any collection of pennies. The article can serve as a framework at least. NP Practitioners can probably give a more informed feedback...

[ EDIT: saw that Baghead had already commented on the article at the time and found it not to be useful... ]